1. Introduction

Highly sensitive photon number resolving detectors (PNRDs), which can determine the number of photons in light pulses, are currently the subject of intense interest [1], particularly in the field of quantum communications [2,3]. Superconducting transition edge sensors (TESs) are very promising candidates due to their high quantum efficiency, excellent photon number resolution and negligible dark count rates [4,5]. The detection mechanism in a TES involves a resistance change in the superconductor due to the absorption of photon energy [6], which is proportional to the number of photons. In order to accurately determine the number of photons in incident light pulses, a high system detection efficiency (DE) is crucial to PNRDs. The system DE is determined by the product of the intrinsic photon absorption coefficient of the TES and the optical coupling efficiency between the incident light and the active area. High photon absorption efficiency can be achieved by the use of an optical cavity [7–9], whereby the TES is formed between an anti-reflection (AR) layer and a high-reflectivity mirror. High optical coupling efficiency can be obtained by the use of a TES device with an active area large enough to enclose the light spot size. The required size of the active area depends on the distance (w _gap) between the fiber end and the device surface, and the mode field diameter (MFD) of an optical fiber. Especially, the large active area is required due to the divergence of the beam size if w _gap is much longer than the Rayleigh length z ₀ (for instance z ₀~23 μm for MFD~5 μm and λ~850 nm). This large active area considerably degrades the performance of TES devices, particularly the photon number resolution in high speed TESs. In addition, due to undesirable optical interference between the fiber end and the device surface, a wavelength dependence of the optical reflectance is observed. This means that the DE also becomes wavelength dependent, which increases the uncertainty when measuring broadband light sources such as squeezed vacuum light and entangled photon pairs. In this paper, we report on a new fiber coupling configuration using a titanium-based TES (Ti-TES), in which a Ti-TES embedded in an optical cavity is directly coupled to an optical fiber by a small index-matched gap with a spacing of less than 1 μm. Ti-TES devices, whose superconducting critical temperature (T _c) is ~300 mK, have already demonstrated a high-speed signal response with a decay time constant of 100 ns and a timing jitter of 30 ns [9], but their photon number resolution is limited compared to extremely low-T _c TES devices. However, with the proposed fiber-coupling method, the active area is reduced to the size of the MFD, which brings about drastic improvements in the photon number resolution without sacrificing the DE or the high response speed. In this paper, we describe the optical and electrical performance of such a fiber-coupled Ti-TES, and demonstrate the highest ever reported DE at a wavelength of approximately 850 nm.

2. Titanium transition edge sensor with optical cavity

The Ti-TES device used in this study comprises a titanium superconductor formed by dc magnetron sputtering in an optical cavity composed of multi-layered dielectric films [9–11]. Figure 1(a) shows an SEM image of a cross-sectional slice through the fabricated Ti-TES produced by focused ion beam milling. From ellipsometry measurements, the complex refractive index of the titanium film at 850 nm is n=3.95+4.13i at room temperature. The cavity consists of multiple dielectric layers of SiO₂ (n=1.48) and Ta₂O₅ (n=2.04) deposited by ion beam assisted sputtering. A total of 7 layers are used for the AR coating, and 15 for the high reflection mirror (Reflectance>99.9%). The thicknesses of these dielectric films are optimized to obtain a wide spectral bandwidth of ~40 nm at the wavelength of interest (λ_eff=850 nm). As shown in Fig. 1(b), a bare optical fiber (outer diameter ~125 μm) without a ferrule is coupled to the cavity from the front side. The fiber core position is aligned to the center of the TES using back-side through-chip imaging with a microscope [11]. To form a robust coupling, the space gap w _gap between the fiber end and the device surface is filled with an ultraviolet-curable resin after the fiber alignment process. The refractive index (n=1.56) of the resin is well matched to that of the fiber core (n~1.44). The spot size required to obtain a 99% photon flux from the optical fiber (MFD~5 μm and numerical aperture~0.14) is estimated to be 8 μm assuming w _gap=1 μm. Thus, the sensitive region of the TES device used here is set to 10 × 10 μm² with a thickness in 22 nm. The fiber alignment error is estimated to be less than 1 μm, and so the coupling loss due to such misalignment is thought to be negligible.

 

Fig. 1 (a) SEM image of a cross section of an optical TES cavity, produced by focused ion beam milling. The dark and bright layers represent SiO₂ and Ta₂O₅, respectively. The Ti-TES is located between the two red arrows. The upper surface is covered with a tungsten (W) layer as protection against damage by the Ga+ ion beam during milling. The inset is an enlarged view of the contact between the TES and Nb electrodes. (b) Cross-sectional schematic showing the optical-fiber coupling. The fiber is aligned under a microscope using back-side through-chip imaging. A gap with a thickness w _gap between the exit end of the fiber and the surface of the antireflection layers is filled with an ultraviolet curable resin.

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3. Results

3.1 Photon absorption efficient

The intrinsic photon absorption coefficient of the fiber-coupled cavity-embedded TES device is analyzed by performing absolute reflectance measurements, and can be deduced from the return loss RL(λ)=P _r/P _i, where P _i is the incident optical power to the device and P _r is the return power from the device back to the light source. In these measurements, we follow the method defined in the international standard of IEEE 61300-3-6, which requires an external-cavity laser source (tunable from 835 to 870 nm), a branching device and three power meters. From the values of RL(λ), the absolute reflectance R(λ) of the TES device is simply given as R(λ) = RL(λ)/ξ, where ξ is the efficiency of the light travelling back to the optical fiber. The value of ξ is determined to be ~82% with a fiber-coupled gold mirror, which has a known reflectance of ~98%. The circles in Fig. 2 are experimental results for R(λ) measured at a device temperature below 1 K. In the figure, the simulated wavelength dependence of the cavity reflectance without fiber coupling (w _gap=∞) and with the fiber-coupled TES for w _gap=λ_eff/4, 1.4×(λ_eff/4), 2×(λ_eff /4), 10 μm, and 30 μm is also shown, predicted by an optical thin film coating simulator, TFCalc (Software Spectra, Inc.). The minimum reflectance value for the cavity design curve with w _gap=∞ is 0.06% (99.94% absorption in TES) at approximately 850 nm. The experimental values are slightly larger, however, and agree well with the simulated reflectance for w _gap=1.4×(λ_eff/4). A gap spacing of 1.4×(λ_eff/4) corresponds to a physical thickness of 300 nm, which is consistent with the step height of 240 nm between the device surface and the niobium wiring, as shown in inset of Fig. 1(a). This step is produced by a lift-off process during niobium wiring fabrication. It should be noted that the measured reflectance is remarkably small at less than 0.5% over the wavelength range from 835 to 870 nm, which implies that the device has a high absorption efficiency of over 99.5% in this range. Besides, no obvious fringes were observed during the reflectance measurements, which imply that the detection efficiency would have a flat spectral response in the observed wavelength range. It is interesting that the reflectance for w _gap=λ_eff/4 and 2×(λ_eff/4) gives the upper- or lower-limit of the intensity of interference fringes, respectively. Thus, the maximum efficiency can be obtained with w _gap=2×(λ_eff/4). A similar efficiency enhancement has been previously reported [12].

 

Fig. 2 The solid lines show simulation results of wavelength dependent reflectance for an optimized cavity and fiber-coupled devices with w_gap values of λ_eff/4, 1.4×(λ_eff/4), 2×(λ_eff/4), 10 μm, 30 μm, and infinity (no fiber coupling). The circles are the measured reflectance results for the fiber-coupled Ti-TES device, which agree well with the simulation results for w _gap=1.4×(λ_eff/4).

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3.2 Detection efficiency

The total system DE for the fiber-coupled TES device is determined by irradiating the device with extremely weak coherent laser pulses with a mean photon number μ_i per pulse. To determine the DE for different values of μ_i, we compare the observed photon detection probability with a Poisson distribution, which is expressed [13] as $P_{Poisson}(n|{\mu }_i)={(\eta {\mu }_i)}^n\exp (-\eta {\mu }_i)/n!$, where η is the system DE and n is a number of the photon state. The observed detection probability is calculated from the pulse height spectra as $P_{TES}(n|{\mu }_i)=S(n)/T$, where S(n) is the summed count for photon state n and T is the total count in the spectrum. The standard deviation in $P_{TES}(n|{\mu }_i)$ is given by $\sqrt{S}/T$, assuming that S obeys Poisson statistics.

Figure 3 shows the experimental setup used to determine the DE of the device. Optical pulses from a laser source with a repetition rate of f~50 kHz are split into two paths by a branching device (BD). One port of the BD is connected to a power meter (D₁) for monitoring the drift of the laser power, and the other is connected to a variable attenuator (VA). The optical power from the VA is measured with a second power meter (D₂). The fiber connector to D₁ has an angled physical contact to prevent back reflection to the BD, while a normal-angle fiber connector is used to D₂ for reliable and polarization-independent measurement of absolute power. The output power from the VA is first measured by D₂. The fiber is then cut and spliced to the TES fiber. The coupling loss due to the splicing process is typically ~0.01 dB. An optical attenuation A is then applied to the average laser power in the VA. In this scheme, μ_i can be described as ${\mu }_i=\frac{\lambda }{hcf}\times \frac{{10}^{-A/10}}{1-{10}^{-R{L}_{D2}/10}}\times P_{D2}$, where P _D2 is the measured power in D₂, RL _D2 is the return loss at the fiber connection to D₂, and h and c are Planck’ constant and speed of light, respectively. The value of the incident mean photon number μ_i must be carefully determined in order to avoid introducing a large uncertainty in the value of DE. Thus, VA and D₂ are carefully calibrated based on national standards [14, 15] at laser wavelengths. The value of RL _D2 is also determined according to IEC 61300-3-6, and we obtain a value of 14.4 dB, corresponding to a reflectance of 3.3%.

 

Fig. 3 Experimental setup for detection efficiency measurement. BD: branching device. VA: variable attenuator. Power meter D₁ is used to monitor the incident laser power. Terminator prevents unwanted back reflection to BD. Power meter D₂ and VA are calibrated based on national standards. A loop near VA is to prevent cladding mode transmission in the optical fiber.

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Figure 4(a) shows the energy distribution observed by a Ti-TES exposed to 844-nm laser pulses with μ_i=1.5. From the peak counts in the distribution, we can determine the values of P _TES(n|μ_i) for photon states n from 0 to 7, μ_i from 3×10⁻³ to 2 per pulse and a total count T of 10⁵. These results are plotted as circles with error bars in Fig. 4(b). By fitting these data sets using P _Poisson(n|μ_i), we obtain η=0.9814±0.0016 at 844 nm. The reduced χ² in the fitting is χ²/ν=0.92 for a degree of freedom ν=41, which implies that the measured errors are consistent with the theoretical errors estimated by the Poisson distribution. A similar experiment carried out at a wavelength of 853 nm gives η=0.9771±0.0037, suggesting that the DE is spectrally flat (<1% variation) in the region around 850 nm. The expanded uncertainty, defined by GUM [16], is estimated to 1% in this detection efficiency measurement. Note that although only one device is discussed in the present paper, measurements carried out on an additional four devices from the same wafer confirm a DE of ~98% with a standard deviation of ~0.3%.

 

Fig. 4 (a) Energy distribution of a pulsed laser source measured using the Ti-TES (circles). The light source has a wavelength of 844 nm and an incident photon number μ_i of ~1.5. Fitted curves using a Poisson distribution convolved with a Gaussian function are also shown (solid). Note that the vertical axes are drawn in log scale. (b) Measured detection probability (circles) of each photon stage for μ_i from 3×10⁻³ to 2 photons/pulse with error bars. Fitted curves using a Poisson distribution are also shown.

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4. Summary

We have demonstrated a high-detection-efficiency photon number resolving detector using titanium-based transition edge sensors optimized for photons at 850 nm wavelengths. The TES device formed in an optical cavity is coupled to an optical fiber by means of an index-matched spacing of ~300 nm. In this configuration, more than 99.5% absorption in the wavelength range from 830 to 870 nm was deduced from the reflectance measurements. The overall detection efficiency was determined to 98±1% at wavelengths of 844 and 853 nm, which is the highest value ever reported for this wavelength range. The energy resolution was 0.42 eV FWHM for a 10×10 μm² active area. With these advantages, such detectors are expected to play an important role in fields such as quantum communications, where control of photon numbers in quantum states is required.